The Stationary wavelet transform (SWT) is a
wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
algorithm designed to overcome the lack of translation-invariance of the
discrete wavelet transform
In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal ...
(DWT). Translation-invariance is achieved by removing the downsamplers and upsamplers in the DWT and upsampling the filter coefficients by a factor of
in the
th level of the algorithm. The SWT is an inherently redundant scheme as the output of each level of SWT contains the same number of samples as the input – so for a decomposition of N levels there is a redundancy of N in the wavelet coefficients. This algorithm is more famously known as "''algorithme à trous''" in French (word ''trous'' means holes in English) which refers to inserting zeros in the filters. It was introduced by Holschneider et al.
Implementation
The following block diagram depicts the digital implementation of SWT.
In the above diagram, filters in each level are up-sampled versions of the previous (see figure below).
KIT
Applications
A few applications of SWT are specified below.
* Signal denoising
* Pattern recognition
* Brain image classification
* Pathological brain detection
[{{cite journal, last1=Dong, first1=Z., title=Magnetic Resonance Brain Image Classification via Stationary Wavelet Transform and Generalized Eigenvalue Proximal Support Vector Machine, journal=Journal of Medical Imaging and Health Informatics, date=2015, volume=5, issue=7, pages=1395–1403, doi=10.1166/jmihi.2015.1542]
Synonyms
* Redundant wavelet transform
* Algorithme à trous
* Quasi-continuous wavelet transform
* Translation invariant wavelet transform
* Shift invariant wavelet transform
* Cycle spinning
* Maximal overlap wavelet transform (MODWT)
* Undecimated wavelet transform (UWT)
See also
*
wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
* wavelet entropy
*
wavelet packet decomposition Originally known as optimal subband tree structuring (SB-TS), also called wavelet packet decomposition (WPD)
(sometimes known as just wavelet packets or subband tree), is a wavelet transform where the discrete-time (sampled) signal is passed through ...
References
Wavelets